A process and an apparatus for building tires for vehicle wheels includes assembling at least one tire component through laying of strip-like elements disposed circumferentially close to each other on a deposition surface radially external to the forming drum, wherein laying of each strip-like element includes: radially moving the strip-like element close to the deposition surface; locking a central portion of the strip-like element against the deposition surface; and pulling the strip-like element in opposite directions starting from the central portion toward the opposite ends thereof, while the central portion is maintained in a locked condition against the deposition surface, so as to progressively lay down the strip-like element against the deposition surface.